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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdmsscn | Structured version Visualization version GIF version |
Description: π is a subset of β. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
dvdmsscn.s | β’ (π β π β {β, β}) |
dvdmsscn.x | β’ (π β π β ((TopOpenββfld) βΎt π)) |
Ref | Expression |
---|---|
dvdmsscn | β’ (π β π β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restsspw 17420 | . . . 4 β’ ((TopOpenββfld) βΎt π) β π« π | |
2 | dvdmsscn.x | . . . 4 β’ (π β π β ((TopOpenββfld) βΎt π)) | |
3 | 1, 2 | sselid 3980 | . . 3 β’ (π β π β π« π) |
4 | elpwi 4613 | . . 3 β’ (π β π« π β π β π) | |
5 | 3, 4 | syl 17 | . 2 β’ (π β π β π) |
6 | dvdmsscn.s | . . 3 β’ (π β π β {β, β}) | |
7 | recnprss 25853 | . . 3 β’ (π β {β, β} β π β β) | |
8 | 6, 7 | syl 17 | . 2 β’ (π β π β β) |
9 | 5, 8 | sstrd 3992 | 1 β’ (π β π β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 β wss 3949 π« cpw 4606 {cpr 4634 βcfv 6553 (class class class)co 7426 βcc 11144 βcr 11145 βΎt crest 17409 TopOpenctopn 17410 βfldccnfld 21286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 ax-resscn 11203 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-rest 17411 |
This theorem is referenced by: dvxpaek 45357 etransclem17 45668 etransclem18 45669 etransclem20 45671 etransclem21 45672 etransclem22 45673 etransclem29 45680 etransclem31 45682 etransclem34 45685 etransclem43 45694 etransclem46 45697 |
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